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Definite Integration question

2014 · Shift 1 · Q26
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  2. /JEE Advanced
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  4. /Definite Integration
  5. /2014 · Shift 1 · Q26

Definite Integration question

2014 · Shift 1 · Q26

JEE AdvancedMathematicsDefinite IntegrationMultiple correct+3 / −1
Let f:(0,∞)→Rf:\left( {0,\infty } \right) \to Rf:(0,∞)→R be given by f(x)f\left( x \right)f(x)= ∫1xxe−(t+1t)tdt\int\limits_{{1 \over x}}^x {{{{e^{ - \left( {t + {1 \over t}} \right)}}} \over t}} dtx1​∫x​te−(t+t1​)​dt. Then
  1. A
    f(x)f(x)f(x) is monotonically increasing on [1,∞)\left[ {1,\infty } \right)[1,∞)
  2. B
    f(x)f(x)f(x) is monotonically decreasing on (0,1)(0,1)(0,1)
  3. C
    f(x)f(x)f(x) +f(1x)=0+ f\left( {{1 \over x}} \right) = 0+f(x1​)=0, for all x∈(0,∞)x \in \left( {0,\infty } \right)x∈(0,∞)
  4. D
    f(2x)f\left( {{2^x}} \right)f(2x) is an odd function of xxx on RRR
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